?tbcon

Estimates the reciprocal of the condition number of a triangular band matrix.

Syntax

FORTRAN 77:

call stbcon( norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info )

call dtbcon( norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info )

call ctbcon( norm, uplo, diag, n, kd, ab, ldab, rcond, work, rwork, info )

call ztbcon( norm, uplo, diag, n, kd, ab, ldab, rcond, work, rwork, info )

FORTRAN 95:

call tbcon( ab, rcond [,uplo] [,diag] [,norm] [,info] )

lapack_int LAPACKE_stbcon( int matrix_order, char norm, char uplo, char diag, lapack_int n, lapack_int kd, const float* ab, lapack_int ldab, float* rcond );

lapack_int LAPACKE_dtbcon( int matrix_order, char norm, char uplo, char diag, lapack_int n, lapack_int kd, const double* ab, lapack_int ldab, double* rcond );

lapack_int LAPACKE_ctbcon( int matrix_order, char norm, char uplo, char diag, lapack_int n, lapack_int kd, const lapack_complex_float* ab, lapack_int ldab, float* rcond );

lapack_int LAPACKE_ztbcon( int matrix_order, char norm, char uplo, char diag, lapack_int n, lapack_int kd, const lapack_complex_double* ab, lapack_int ldab, double* rcond );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.h

Description

The routine estimates the reciprocal of the condition number of a triangular band matrix A in either the 1-norm or infinity-norm:

κ₁(A) =||A||₁ ||A^-1||₁ = κ_∞(A^T) = κ_∞(A^H)

κ_∞(A) =||A||_∞ ||A^-1||_∞ =κ₁ (A^T) = κ₁(A^H) .

Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

norm	CHARACTER*1. Must be '1' or 'O' or 'I'. If `norm` = '1' or 'O', then the routine estimates the condition number of matrix `A` in 1-norm. If `norm = 'I'`, then the routine estimates the condition number of matrix `A` in infinity-norm.
uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates whether `A` is upper or lower triangular: If `uplo = 'U'`, the array `ap` stores the upper triangle of `A` in packed form. If `uplo = 'L'`, the array `ap` stores the lower triangle of `A` in packed form.
diag	CHARACTER*1. Must be 'N' or 'U'. If `diag = 'N'`, then `A` is not a unit triangular matrix. If `diag = 'U'`, then `A` is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array `ab`.
n	INTEGER. The order of the matrix `A`; `n` ≥ 0.
kd	INTEGER. The number of superdiagonals or subdiagonals in the matrix `A`; `kd` ≥ 0.
ab, work	REAL for stbcon DOUBLE PRECISION for dtbcon COMPLEX for ctbcon DOUBLE COMPLEX for ztbcon. Arrays: `ab(ldab,)`, `work()`. The array `ab` contains the band matrix `A`. The second dimension of `ab` must be at least `max(1,n)`. The array `work` is a workspace for the routine. The dimension of `work` must be at least `max(1, 3n)` for real flavors and `max(1, 2n)` for complex flavors.
ldab	INTEGER. The leading dimension of the array `ab`. (`ldab` ≥ `kd` +1).
iwork	INTEGER. Workspace array, DIMENSION at least `max(1, n)`.
rwork	REAL for ctbcon DOUBLE PRECISION for ztbcon. Workspace array, DIMENSION at least `max(1, n)`.

Output Parameters

rcond

REAL for single precision flavors.

DOUBLE PRECISION for double precision flavors.

An estimate of the reciprocal of the condition number. The routine sets rcond =0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime rcond is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular.

info

INTEGER. If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine tbcon interface are as follows:

ab	Holds the array `A` of size `(kd+1,n)`.
norm	Must be '1', 'O', or 'I'. The default value is '1'.
uplo	Must be 'U' or 'L'. The default value is 'U'.
diag	Must be 'N' or 'U'. The default value is 'N'.

Application Notes

The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10r. A call to this routine involves solving a number of systems of linear equations A*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2*n(kd + 1) floating-point operations for real flavors and 8*n(kd + 1) operations for complex flavors.